Representation theorem for interlaced q-bilattices

Yu.M.Movsisyan, D.S.Davidova

Department of mathematics and mechanics

Yerevan State University, Yerevan, Armenia

E-mail: yurimovsisyan@yahoo.com

Annotation. It is proved in this work that every interlaced q-bilattice is isomorphicto the superproduct of two q-lattices.

Introduction and preliminaries. Bilattices are the algebraic structures that wereintroduced by Ginsberg [1, 2] as a general and uniform framework for a diversity of appli-cations in arti�cial intelligence. In a series of papers it was shown that these structurescan serve as a foundation for many areas, such as logic programming [3-5], computationallinguistics, distributive knowledge processing and reasoning with imprecise information.Bilattices are useful in the context of fuzzy logics as well.

A bilattice is the algebra, (L;∧,∨, ∗,4), with four binary operations such that thefollowing two reducts, L1 = (L;∧,∨) and L2 = (L; ∗,4), are lattices.

A bilattice is called interlaced if all the basic bilattice operations are order preservingwith respect to the both corresponding orders.

In papers [1, 3, 5-9 ] bounded distributive or bounded interlaced bilattices are char-acterized (note that distributive lattices with third additional operation are studied in[10]-[13]). In [14], interlaced bilattices without bounds are characterized (see also [15]).

De�nition 1. The algebra, (L;∧), is called a q-semilattice, if it satis�es the followingidentities:

1. a ∧ b = b ∧ a (commutativity);2. a ∧ (b ∧ c) = (a ∧ b) ∧ c (associativity);3. a ∧ (b ∧ b) = a ∧ b (weak idempotency).De�nition 2. The algebra, (L;∧,∨), is called a q-lattice (see [16]), if the reducts, (L;∧)

and (L;∨), are q-semilattices and the following identities, a∧(b∨a) = a∧a, a∨(b∧a) = a∨a(weak absorption), a ∧ a = a ∨ a (equalization) are valid.

For each q-semilattice, (L;∧), there exists a quasiorder, Q (i.e. a re�exive and transitiverelation), which is de�ned in the following manner: aQb↔ a∧b = a∧a. For each q-lattice,(L;∧,∨), we have: aQb↔ a ∧ b = a ∧ a↔ a ∨ b = b ∨ b.

For example, (Z \ {0};∧,∨), where x ∧ y = |(x, y)| and x ∨ y = |[x, y]| (here (x, y) and[x, y] are the greatest common division (gcd) and the least common multiple (lcm) of xand y), is a q-lattice, since x

∧x 6= x and x

∨x 6= x.

De�nition 3. A q-bilattice is an algebraic structure, (L;∧,∨, ∗,4), with the twoq-lattice reducts, L1 = (L;∧,∨) and L2 = (L; ∗,4), which also satis�es the followingidentity: a ∗ a = a ∧ a (the quasiorder of the �rst reduct, (L;∧,∨), is denoted by ≤∧, andthat of the second reduct - by ≤∗).

De�nition 4. The operation, ∗, of the q-semilattice, (L; ∗), is called interlaced with theoperations, ∧ and ∨, of the q-lattice, (L;∧,∨), if the q-semilattice operation, ∗, preservesthe q-lattice quasiorder ≤∧, and q-lattice operations, ∧ and ∨, preserve the q-semilatticequasiorder ≤∗.

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De�nition 5. The q-bilattice, (L;∧,∨, ∗,4), is called interlaced if all the basic q-bilattice operations are quasiorder preserving with respect to the both corresponding qua-siorders.

Let us recall that a hyperidentity is a second order formula of the following type:

∀X1, ..., Xm ∀x1, ..., xn(w1 = w2),

where X1, ..., Xm are functional variables, and x1, ..., xn are objective variables in the words(terms) of w1, w2. Hyperidentities are usually written without the quanti�ers, that is tosay, w1 = w2. We say that in the algebra, (Q;F ), the hyperidentity, w1 = w2, is satis�edif this equality is valid, when every objective variable and every functional variable in itis replaced by any element of Q and by any operation of the corresponding arity from Frespectively (supposing the possibility of such replacement)[17-18].

On characterization of hyperidentities of varieties of lattices, modular lattices, distrib-utive lattices, Boolean and De Morgan algebras see [19-22]. About hyperidentities in term(polynomial) algebras of lattices see [23].

For example, the q-bilattice, L = (L;∧,∨, ∗,4), is interlaced i� it satis�es the followinghyperidentity:

X(Y (X(x, y), z), Y (y, z)) = Y (X(x, y), z).

For a categorical de�nition of hyperidentities, in [17] the (bi)homomorphisms betweenthe two algebras, (Q;F ) and (Q′;F ′), are de�ned as the pair, (ϕ, ψ̃), of the maps:

ϕ : Q→ Q′, ψ̃ : F → F ′, |A| = |ψ̃A|,

with the following condition:

ϕA(a1, ..., an) = (ψ̃A)(ϕa1, ..., ϕan),

for any A ∈ F, |A| = n, a1, ..., an ∈ Q .Algebras with their (bi)homomorphisms, (ϕ, ψ̃), (as morphisms) form a category with

products. The product in this category is called superproduct of algebras and is denoted byQ ./ Q′ for the two algebras, Q and Q′. For example, a superproduct of the two q-lattices,Q(+, ·) and Q′(+, ·), is the binary algebra, Q × Q′ ((+,+), (·, ·), (+, ·), (·,+)), with fourbinary operations, where the pairs of the operations operate componentwise, i.e.

(A,B) ((x, y), (u, v)) = (A(x, u), B(y, v)) ,

and Q ./ Q′ is an interlaced q-bilattice.De�nition 6. The subset, F ⊆ L, is called a �lter of the q-bilattice, (L;∧,∨, ∗,4), if

F satis�es the following conditions:(ff1) if x, y ∈ F, then x ∧ y ∈ F ;(ff2) if x ∈ F, y ∈ L and x ≤∧ y, then y ∨ y ∈ F ;(ff3) if x, y ∈ F, then x ∗ y ∈ F ;(ff4) if x ∈ F, y ∈ L and x ≤∗ y, then y4y ∈ F.Denote the set of all �lters of the q-bilattice, L, by FF (L).De�nition 7. The subset, I ⊆ L, is called an ideal of the q-bilattice, (L;∧,∨, ∗,4), if

I satis�es the following conditions:(fi1) if x, y ∈ I, then x ∧ y ∈ I;(fi2) if x ∈ I, y ∈ L and x ≤∧ y, then y ∨ y ∈ I;(fi3) if x, y ∈ I, then x4y ∈ I;(fi4) if y ∈ I, x ∈ L and x ≤∗ y, then x ∗ x ∈ I.2

Denote the set of all ideals of the q-bilattice, L, by FI(L).Let for each a ∈ L, Bf(a) = {X ∈ FF (L)|a ∈ X}, Bi(a) = {X ∈ FI(L)|a ∈ X}. Let

us de�ne on the sets, Bf(L) = {Bf(a)|a ∈ L} and Bi(L) = {Bi(a)|a ∈ L}, the binaryoperations, ∩∗ and ∪∗, in the following manner:

Bf(a) ∩∗ Bf(b) = {F ∈ FF (L)|(∃X ∈ Bf(a ∧ a)) (∃Y ∈ Bf(b ∧ b))X ∪ Y ⊆ F};Bf(a) ∪∗ Bf(b) = {F ∈ FF (L)|(∃X ∈ Bf(a ∧ a)) (∃Y ∈ Bf(b ∧ b))X ∩ Y ⊆ F};Bi(a) ∩∗ Bi(b) = {F ∈ FI(L)|(∃X ∈ Bi(a ∧ a)) (∃Y ∈ Bi(b ∧ b))X ∪ Y ⊆ F};Bi(a) ∪∗ Bi(b) = {F ∈ FI(L)|(∃X ∈ Bi(a ∧ a)) (∃Y ∈ Bi(b ∧ b))X ∩ Y ⊆ F}.

It is easy to show that (Bf(L);∩∗,∪∗) and (Bi(L);∩∗,∪∗) are q-lattices.

Main result

Theorem. Every interlaced q-bilattice L = (L;∧,∨, ∗,4) is isomorphic to the super-product of the following two q-latticesBf(L) = (Bf(L);∩∗,∪∗) andBi(L) = (Bi(L);∩∗,∪∗),i.e.

L ∼= Bf(L) ./ Bi(L).

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